Allowable axial loads for tubular structures are available in tank codes, AISC-ASD, perhaps ASME B&PV, and a number of other sources, if your design code doesn't specify anything.
If you have significant earth loads, that will reduce the allowable axial load.
You won't have euler (column) buckling since the pipe is laterally braced by the earth. However, you can get shell buckling. ASME piping codes typically refer to the Boiler and Pressure Vessel Code, Section VIII for rules for evaluation of shell buckling.
You might want to have a look at API RP 1102, Road & Railway Crossing Design; and then do the calculations assuming the load from the road or railway is zero (i.e. only loads are internal pressure, thermal expansion and applied earth loads). The code has a very easy, step by step claculation, and gives typical design numbers for different types of earth and backfilling techniques etc. I suspect that as the client is using a pipeline design code, they'd be familliar with API RP 1102, and would accept and results from this calculation.
Rules for railroad crossing would deal with soil load, which is a hoop effect. The question is axial buckling, so I presume that is what the customer is asking for.
Use Section VIII Div 1 rules for cylinders under axial compression or Code Case 2286.
I believe some of the AWWA pipe codes and manuals also give information for buckling from soil pressure. Which, as indicated above, is not axial loading, but uneven external loading. I assume axial loading, if present, would be from temperature changes, soil movement, or seismic movement in a restrained pipe.
Note that for small-diameter pipe, application of the local buckling criteria will just show that the pipe can yield before it buckles. And if it's large enough and thin enough for that not to be the case, then earth loading is liable to be more significant.
The operating temperature is around 300F. I have large deflections and possibly large axial forces. The pipeline is restrained axially by friction and sometimes by natural or real anchors.
If the axial forces are high the pipe can buckle. The soil will act against the movement and it will reduce the vertical and/or lateral movement.
In what circumstances the vertical and/or lateral buckling can occur? How can I calculate the maximum axial force to prevent the buckling considering the elastic-plastic behavior of the soil?
Assuming the backfill is normal compacted backfill, I don't believe lateral buckling would be an issue.
You might check with people familiar with piling. Piles are normally designed assuming they have lateral support from surrounding soil, not as free-standing columns, and I suppose similar issues arise there.
See page 27 of the document. It doesn't specifically address axial buckling, but rather bending at ells caused by axial thermal expansion. Also see the appendix referenced by this section.
If you are lookng at column buckling, i.e. Euler buckling, you will find that the forces required to prevent it are very low, so the soil will prevent column buckling. Again, shell buckling (like when you overload a soda can by stepping on it) is another matter and the soil will not prevent that.
If you want to do a calculation for column buckling, you can run a FEA with a long beam with periodic spring restraints representing the soil stiffness. Look to the B31.1 appendix VII for a method to calculate effective springs that simulate soil stiffness. But you should learn from this exercise that there is no problem with Euler buckling of buried pipe.
Of course there are local effects such as wrinkling due to bending a pipe over a change in direction, buckling due to soil loads under some circumstance, buckling due to external pressure (e.g. pipe under the water table), etc., but my understanding is that the question is with respect to axial loads.
The pipeline is miles and miles in length. The pipe size is 8". No questions about shell buckling. It will not happen.
I know the soil characteristics and the spring rate for interaction with the pipe.
The level of stresses is under allowable but the axial buckling happened and the pipe popped out from the ground. I guess that the cause is axial buckling
It is unlikely that axial buckling caused it unless the pipe burial is very shallow.
What is more likely is that you had a vertical change in direction, so it is the thermal expansion forces pushing on that change in direction that pushed the pipe out of the ground. Folks use intermediate anchors at such changes in direction to handle that effect.
In fact this is also my opinion. It is more likely that the vertical changes in direction combined with large thermal expansion cause the problem.
But I still need to prove that Euler buckling did not happened.
It is easier to do it the other way (if you have the profile of the piping). Given a change in pipe slope, you can calculate the force required to offset the forces due to thermal expansion. This is a simple force vector problem. Then look at what there was to try to restrain the pipe.
Look at B31.1 Appendix VII to figure where the virtual anchors are, and how much the pipe would actually move should it be able to.
See if all this explains what was observed.
Note that if the pipe is perfectly straight, it takes negligable lateral restrain to prevent axial buckling.
Dan511- This sounds more like classic upheaval buckling than axial buckling. Upheaval buckling is a real problem in offshore pipelines due to the difference between operating & installation temperatures, the difficulty of providing a smooth pipeline profile and the difficulty of trenching and backfilling offshore lines. But the lessons learned form offshore are clearly applicable onshore!
A lot of work has been done on this, especially in the late 80s. For example SPE 68224 concerns a HP/HT gas line in Abu Dhabi, where section of the line started to protude from the trench soon after operations started; there are lots of OTC papers: OTC 6846 "Soil response for upheaval buckling analysis, full scale lab tests", OTC6333 "FEA model for analysing upheaval buckling response of submarine pipelines", OTC 6335 "Design of submarine pipelines against upheaval buckling", OTC 6488 "Upheaval buckling failures of insulated buried pipelines: a case history" etc etc.
